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The all-pay auction with complete information and identity-dependent externalities

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Abstract

We derive a necessary and sufficient condition for the existence of equilibria with only two active players in the all-pay auction with complete information and identity-dependent externalities. This condition shows that the generic equilibrium of the standard all-pay auction is robust to the introduction of “small” identity-dependent externalities. In general, however, the presence of identity-dependent externalities invalidates well-established qualitative results concerning the set of equilibria of the first-price all-pay auction with complete information. With identity-dependent externalities, equilibria are generally not payoff equivalent, and identical players may earn different payoffs in equilibrium. These observations show that Siegel’s (Econometrica 77(1), 71–92, 2009) results characterizing the set of equilibrium payoffs in all-pay contests, including the all-pay auction as a special case, do not extend to environments with identity-dependent externalities. We further compare the all-pay auction with identity-dependent externalities to the first-price winner-pay auction with identity-dependent externalities. We demonstrate that the equilibrium payoffs of the all-pay auction and the “undominated strategy equilibrium” payoffs of the winner-pay auction (Funk in Int J Game Theory 25(1), 51–64, 1996) cannot be ranked unambiguously in the presence of identity-dependent externalities by providing examples of environments where equilibrium payoffs in the all-pay auction dominate those of the undominated strategy equilibria in the winner-pay auction and vice versa.

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Notes

  1. Recent applications of the all-pay auction include Baye et al. (2012), Chatterjee et al. (2012), Groh et al. (2012), Iaryczower and Mattozzi (2012), Jordan and Meirowitz (2012), Konrad and Kovenock (2012), Levy and Razin (2012), Pastine and Pastine (2012a, b) and Roberson and Kvasov (2012).

  2. See for example das Varma (2002) for an analysis of standard open and sealed bid winner-pay auctions with IDE and Jehiel et al. (1999) or Aseff and Chade (2008) for optimal mechanism design in the presence of identity-dependent externalities. Biran and Forges (2011) show how identity-dependent externalities can explain collusive rings in auctions. The effects of identity-dependent externalities in bargaining and negotiation games are analyzed in Jehiel and Moldovanu (1995a) and Jehiel and Moldovanu (1995b). Jehiel and Moldovanu (1999) and Cornet (2000) consider markets with resale in the presence of identity-dependent externalities.

  3. Funk’s (1996) equilibrium concept is based on a game in which players simultaneously bid and the seller deterministically chooses a winner from the set of high bidders.

  4. Note that when externalities are identity-independent, i.e., \(v_{ij}=v_{ik}\) for all players \(i\in I\) and all \(j,k\ne i\), a player’s valuation only depends on the state of winning or losing the prize, but not on the allocation of the prize given the event that he loses. All of the Baye et al. (1996) results, and Siegel’s (2009) payoff characterization, apply in this case.

  5. Although we take bid costs to be linear, our analysis also applies directly to an environment where players have simple cost functions for bidding that take the form \(c_i(x_i)=\gamma _iC(x_i),\) where \(\gamma _i>0\) for every \(i\in I\) and \(C(x_i)\) is continuous, strictly increasing, and unbounded with \(C(0)=0\) (see Moldovanu and Sela 2001; Siegel 2010). In this case, taking a standard transformation of each player \(i\)’s utility by dividing by \(\gamma _i\) and then treating the normalized cost \(z_i=C(x_i)\) as a bid, the same analysis could be carried out on the transformed valuations \(\tilde{v}_{ij}\equiv v_{ij}/\gamma _i\) and bids \(z_i\) (subject to inverting the transformation to make statements about payoffs).

  6. Assumption 1 implies that reaches are strictly positive and thus rules out the existence of a pure strategy equilibrium in which all players bid zero and the outcome is determined by the tie-breaking rule.

  7. When constrained to the special case of the single-prize all-pay auction, in Siegel’s (2009) terminology, the power of a player is his valuation of winning at the second highest reach.

  8. In the context of a single-prize all-pay contest, Siegel’s (2009) notion of generic refers to the case that (i) only one player has power 0 and (ii) this player’s valuation of winning is strictly decreasing at his reach. Note that the second condition is always satisfied in all-pay auctions where the cost is equal to the bid.

  9. Though, because of the use of general cost functions, neither Alcalde and Dahm’s (2010) condition that only two active bidders participate nor their condition on the expected bid of each player necessarily hold in the equilibria identified by Siegel (2009).

  10. In the case of the all-pay auction with IDE, the sharing rule is already uniquely determined for all strategy profiles which exhibit a unique highest bid. The part of the sharing rule which will be determined endogenously is concerned with ties. Therefore, we can say that in the game \(\varGamma ^\mathrm{AP}\), Simon and Zame’s (1990) result shows that an endogenously determined tie-breaking rule exists, such that an equilibrium in mixed strategies exists in the game with this tie-breaking rule.

  11. A number of other results in the literature on existence of equilibria in discontinuous games can be applied, although verification that the stated sufficient conditions are satisfied is sometimes taxing. For instance, with considerable work, one may show that the finite deviation property holds in the mixed extension of the all-pay auction with identity-dependent externalities, so that Theorem 2.9 of Reny (2009) applies. It is useful to note, however, that several existence theorems appearing in the literature, which cite the standard all-pay auction as an example of the theorem’s applicability, do not apply generally to all-pay auctions with identity-dependent externalities. For example, neither the uniform payoff security condition of Monteiro and Page Jr. (2007) nor the uniform diagonal security condition of Prokopovych and Yannelis (2014) applies generally to all-pay auctions with identity-dependent externalities.

  12. As a semi-metric, \(d(i,j)\equiv r_{ij}\) is required to satisfy nonnegativity, identity of indiscernables (\(r_{ij}=0\) if and only if \(i=j\)), and symmetry (\(r_{ij}=r_{ji}\)). It is not required to satisfy the triangle inequality. Note that, from Assumption 1, the first two conditions are always satisfied.

  13. A similar configuration of reaches is examined in Klose and Kovenock’s (2012) analysis of extremism and moderation, representing the case of two centrist players and one radical.

  14. Even if such a tie-breaking rule is not employed, equilibria with the same payoffs exist if players are allowed to play mixed strategies.

  15. Funk (1996) examines a sequential version of \(\varGamma ^\mathrm{WP}\) in which the \(n\) bidders submit their bids simultaneously in a first stage and an additional player, the seller, chooses in the second stage the winner from the set of high bidders. All players are constrained to play pure strategies, and bids must lie in the closure of the set of undominated strategies.

  16. Different elements of the set of undominated strategy equilibrium winners and bids may be supported by different tie-breaking rules. In particular, if there exist players \(i,j\in I\), \(i\ne j\) such that \(r_{ij}=r_{ji}\ge \max \{\gamma _i,\gamma _j\},\) then there exist two different deterministic tie-breaking rules, one under which \((i,r_{ij})\) is in the set of equilibrium winners and winning bids and another one under which \((j,r_{ij})\) belongs to this set.

  17. Although it is easy to rule out mixed-strategy equilibria (with strategies in the closure of the set of weakly undominated bids) having payoffs other than the undominated strategy equilibrium payoffs in the standard winner-pay auction, these arguments do not appear to carry over to the case of winner-pay auctions with IDEs. A full characterization of the set of all winner-pay equilibrium payoffs in pure or mixed strategies is beyond the scope of the paper.

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Authors and Affiliations

  1. University of Zurich, 8006, Zurich, Switzerland

    Bettina Klose

  2. Chapman University, Orange, CA, 92866, USA

    Dan Kovenock

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  1. Bettina Klose
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Correspondence to Dan Kovenock.

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We have benefited from helpful conversations with Kai Konrad, Wolfgang Leininger, Benny Moldovanu, and Ron Siegel and the comments of two anonymous referees and seminar participants at Chapman University, CORE (Université Catholique de Louvain), Erasmus University Rotterdam, the Max Planck Institute for Tax Law and Public Finance, Mississippi State University, Purdue University, the University of Bonn, the University of California at Riverside, the University of York, and the University of Zurich. Earlier versions of this paper were presented at the 2011 Australasian Economic Theory Workshop in Adelaide, the 2011 Meetings of the European Economic Association and Econometric Society in Oslo, the 2011 Society for the Advancement in Economic Theory Conference in Faro, and the 2012 Conference on Contests, Mechanisms, and Experiments at the University of Exeter. Bettina Klose gratefully acknowledges the financial support of the European Research Council (ERC Advanced Investigator Grant, ESEI-249433) and the Swiss National Science Foundation (SNSF 100014 135257). Dan Kovenock has benefited from the financial support of the Social Science Research Center Berlin (WZB) and the Max Planck Institute for Tax Law and Public Finance in Munich.

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Klose, B., Kovenock, D. The all-pay auction with complete information and identity-dependent externalities. Econ Theory 59, 1–19 (2015). https://doi.org/10.1007/s00199-014-0848-5

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